Characteristic implies normal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
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To learn more about the similarities and differences between characteristicity and normality, refer the survey article Characteristic versus normal

Statement

Verbal statement

Every characteristic subgroup of a group is a normal subgroup.

Symbolic statement

Let $H$ be a characteristic subgroup of $G$. Then, $H$ is normal in $G$.

Property-theoretic statement

The subgroup property of being characteristic is stronger than the subgroup property of being normal.

Definitions used

Characteristic subgroup

Further information: Characteristic subgroup

The definitions we use here are as follows:

• Hands-on definition: A subgroup $H$ of a group $G$ is termed a characteristic subgroup, if for any automorphism $\sigma$ of $G$, we have $\sigma (H)=H$.
• Definition using function restriction expression: We can write characteristicity as the invariance property with respect to automorphisms:

Characteristic = Automorphism $\to$ Function

This is interpreted as: any automorphism from the whole group to itself, restricts to a function from the subgroup to itself. In other words, the subgroup is invariant under automorphisms.

• Definition using relation implication expression: We can write characteristicity as:

Automorphic subgroups $\implies$ Equal

In other words, any subgroup obtained by taking the image of this subgroup under an automorphism of the whole group, must be equal to it.

Definitions for normal

Further information: Normal subgroup

The definitions we use here are as follows:

• Hands-on definition: A subgroup $H$ of a group $G$ is termed normal, if for any $g\in G$, the inner automorphism $c_{g}$ defined by conjugation by $g$, namely the map $x\mapsto gxg^{-1}$, gives an isomorphism on $H$. In other words, for any $g\in G$:

$c_{g}(H)=H$

or more explicitly:

$gHg^{-1}=H$

Implicit in this definition is the fact that $c_{g}$ is an automorphism. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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• Definition using function restriction expression: We can write normality as the invariance property with respect to inner automorphisms:

Normal = Inner automorphism $\to$ Function

In other words, any inner automorphism on the whole group restricts to a function from the subgroup to itself.

• Definition using relation implication expression: We can write:

Normal = Conjugate subgroups $\implies$ Equal

In other words, any subgroup conjugate to the given one, must be equal to it.

Proof

Hands-on proof

Given: Let $H$ be a characteristic subgroup of $G$. Then, for any automorphism $\sigma$ of $G$, $\sigma (H)=H$. Our goal is to show that for any $g\in G$, $gHg^{-1}=H$.

To prove: For any $g\in G$, for $c_{g}$ denotes conjugation by $g$ i.e. the map $x\mapsto gxg^{-1}$.

Proof: $c_{g}$ is an inner automorphism, so it is an automorphism. Thus, invoking characteristicity, we have $c_{g}(H)=H$, i.e. $gHg^{-1}=H$.

Thus $H$ is a normal subgroup of $G$.

Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic

Normality is the invariance property with respect to inner automorphisms, and characteristicity is the invariance property with respect to automorphisms. Explicitly:

Normal = Inner automorphism $\to$ Function

Characteristic = Automorphism $\to$ Function

Since the left side of normality implies the left side of characteristicity, every characteristic subgroup is normal.

Using relation implication expressions

This subgroup property implication can be proved using relation implication expressions for the subgroup properties
View other implications proved in this way OR Read a survey article on the topic

A subgroup is normal if any subgroup that resembles it as a conjugate is the same as it; a subgroup is characteristic if any subgroup that resembles it via an automorphism is the same as it. Since the notion of resembling as a conjugate is stronger than the notion of resembling via an automorphism, every characteristic subgroup is normal.

In terms of equivalence classes of elements

A normal subgroup is a subgroup that is a union of conjugacy classes; a characteristic subgroup is a subgroup that is a union of automorphism classes. Since every automorphism class is a union of conjugacy classes, every characteristic subgroup is normal.

Converse

Normal subgroups need not be characteristic

Further information: Normal not implies characteristic

Normal-to-characteristic

A property that, along with normality, implies characterisicity, is termed a normal-to-characteristic subgroup property. A typical example is the property of being automorph-conjugate. This property plays on the resemblance-based definitions of normality and characteristicity.

Groups where every normal subgroup is characteristic

A group in which the normal subgroups are precisely the same as the characteristic subgroups, is termed a N=C-group.

Intermediate properties

Intermediate invariance properties

The invariance property for an automorphism property that is weaker than the property of being an inner automorphism, lies somewhere between characteristicity and normality. Here are some examples:

• Extensible automorphism-invariant subgroup is a subgroup that is invariant under every extensible automorphism (a property of automorphisms that is weaker than the property of being an inner automorphism)
• Monomial automorphism-invariant subgroup is a subgroup that is invariant under every monomial automorphism

Properties involving additional structure

Invariance properties with respect to automorphisms that preserve additional structure imposed on the group, lie between characteristicity and normality. For instance:

• Topologically characteristic subgroup in a topological group

Other intermediate properties

• Potentially characteristic subgroup is a subgroup that is characteristic in some group containing the bigger group.
• Strongly potentially characteristic subgroup is a subgroup such that there is a group containing the bigger group in which both of them are characteristic.

References

Textbook references

• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info}, Page 17
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}, Page 135, Page 137